Mestre ten-root construction, roots 0,5,9,21,27,31,46,55,64,80: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and this the curve submitted is itsthe minimal Weierstrass model of its Jacobian. Found by enumerating all integer ten-root sets 0 = a_1 < ... < a_10 = R with 64 < R <= 80 (about 1.0e11 sets up to reflection and scaling) and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.
Mestre ten-root construction, roots 0,5,9,21,27,31,46,55,64,80: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and this is its minimal Weierstrass model. Found by enumerating all integer ten-root sets 0 = a_1 < ... < a_10 = R with 64 < R <= 80 (about 1.0e11 sets up to reflection and scaling) and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.
Michael Rubinstein ·
thisthe curve submitted isitsthe minimal Weierstrass model of its Jacobian. Found by enumeratingallinteger ten-root sets 0 = a_1 < ... < a_10= R with 64 < R <= 80 (about 1.0e11 sets up to reflection and scaling)and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.Michael Rubinstein ·